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With COVID-19 lingering around longer than a bad smell, many mathematical models are continually being analyzed to help forecast the future of the pandemic, at local, national, and international levels. This pandemic is far trickier than anyone had predicted and has features that no single model has yet been able to pinpoint. Though the SIR Model (Susceptible-Infected-Removed) is the system of three differential equations that most Center for Disease Control and Prevention (CDC) simulations start off with, we will consider a simpler variation of this more complex, nonlinear model (the SIR model is unsolvable with exact techniques). In our model, we assume the following: there are two compartments - a person is either susceptible or infected. We assume that those who are susceptible can become infected and we assume infected individuals can become susceptible again (once they are recovered from the disease). Define: + bi as-bi where s-proportion of the population susceptible i-proportion of the population infected 4,6> 0 are some real constants t-time since outbreak observed, days Notice thats and i are proportions. This means those values are allowed to be between 0 (0%) and 1.00 (100%), Preliminary Analysis: 1. Draw a compartmental diagram (cascading tanks) showing how the two groups interact 2. Explain whata and bare measures of State their units. 3. Notice that +-0. Explain why this is and what assumption it makes about the population being considered. 4. The point (s)-(0.0) is always an equilibrium point. Verify this is true. Why does this make sense?

With COVID-19 lingering around longer than a bad smell, many mathematical models are continually being analyzed to help forecast the future of the pandemic, at local, national, and international levels. This pandemic is far trickier than anyone had predicted and has features that no single model has yet been able to pinpoint. Though the SIR Model (Susceptible-Infected-Removed) is the system of three differential equations that most Center for Disease Control and Prevention (CDC) simulations start off with, we will consider a simpler variation of this more complex, nonlinear model (the SIR model is unsolvable with exact techniques). In our model, we assume the following: there are two compartments - a person is either susceptible or infected. We assume that those who are susceptible can become infected and we assume infected individuals can become susceptible again (once they are recovered from the disease). Define: + bi as-bi where s-proportion of the population susceptible i-proportion of the population infected 4,6> 0 are some real constants t-time since outbreak observed, days Notice thats and i are proportions. This means those values are allowed to be between 0 (0%) and 1.00 (100%), Preliminary Analysis: 1. Draw a compartmental diagram (cascading tanks) showing how the two groups interact 2. Explain whata and bare measures of State their units. 3. Notice that +-0. Explain why this is and what assumption it makes about the population being considered. 4. The point (s)-(0.0) is always an equilibrium point. Verify this is true. Why does this make sense?

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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With COVID-19 lingering around longer than a bad smell, many mathematical models are continually being analyzed to help forecast the future of the pandemic, at local, national, and international levels. This pandemic is far trickier than anyone had predicted and has features that no single model has yet been able to pinpoint. Though the SIR Model (Susceptible-Infected-Removed) is the system of three differential equations that most Center for Disease Control and Prevention (CDC) simulations start off with, we will consider a simpler variation of this more complex, nonlinear model (the SIR model is unsolvable with exact techniques). In our model, we assume the following: there are two compartments - a person is either susceptible or infected. We assume that those who are susceptible can become infected and we assume infected individuals can become susceptible again (once they are recovered from the disease). Define: -as + bi -as-bi where 8- proportion of the population susceptible i-proportion of the population infected 4,6> 0 are some real constants t-time since outbreak observed, days Notice thats and i are proportions. This means those values are allowed to be between 0 (0%) and 1.00 (100%) Preliminary Analysis: 1. Draw a compartmental diagram (cascading tanks) showing how the two groups interact. 2. Explain what a and bare measures of. State their units. 3. Notice that += 0. Explain why this is and what assumption it makes about the population being considered. 4. The point (si) - (0,0) is always an equilibrium point. Verify this is true. Why does this make sense? Model Investigation Going forward, assume that 90% of the population is initially susceptible and 10% is initially infected. 5. Using technology, find the eigenvalues and eigenvectors of this general system. Your solution will depend on a and on b. Note that Wolfram Alpha is completely capable of finding eigenvalues and eigenvectors based on unknown matrix values, a and b. Write the general solution for s(t) and i(t). 6. Let a-b-0.1. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 7. Let a-0.1 and b-0.2. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 8. Let a-0.2 and b-0.1. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 9. Comment on the differences among the last three scenarios. Why does this make practical sense?
Transcribed Image Text:With COVID-19 lingering around longer than a bad smell, many mathematical models are continually being analyzed to help forecast the future of the pandemic, at local, national, and international levels. This pandemic is far trickier than anyone had predicted and has features that no single model has yet been able to pinpoint. Though the SIR Model (Susceptible-Infected-Removed) is the system of three differential equations that most Center for Disease Control and Prevention (CDC) simulations start off with, we will consider a simpler variation of this more complex, nonlinear model (the SIR model is unsolvable with exact techniques). In our model, we assume the following: there are two compartments - a person is either susceptible or infected. We assume that those who are susceptible can become infected and we assume infected individuals can become susceptible again (once they are recovered from the disease). Define: -as + bi -as-bi where 8- proportion of the population susceptible i-proportion of the population infected 4,6> 0 are some real constants t-time since outbreak observed, days Notice thats and i are proportions. This means those values are allowed to be between 0 (0%) and 1.00 (100%) Preliminary Analysis: 1. Draw a compartmental diagram (cascading tanks) showing how the two groups interact. 2. Explain what a and bare measures of. State their units. 3. Notice that += 0. Explain why this is and what assumption it makes about the population being considered. 4. The point (si) - (0,0) is always an equilibrium point. Verify this is true. Why does this make sense? Model Investigation Going forward, assume that 90% of the population is initially susceptible and 10% is initially infected. 5. Using technology, find the eigenvalues and eigenvectors of this general system. Your solution will depend on a and on b. Note that Wolfram Alpha is completely capable of finding eigenvalues and eigenvectors based on unknown matrix values, a and b. Write the general solution for s(t) and i(t). 6. Let a-b-0.1. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 7. Let a-0.1 and b-0.2. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 8. Let a-0.2 and b-0.1. Determine the particular solution for this system and provide a graph displaying s(t) and it). Describe what is happening in the long-run. 9. Comment on the differences among the last three scenarios. Why does this make practical sense?
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